سیاست های بهینه مصرف / سرمایه گذاری با ریسک درآمد غیرمتنوع و محدودیت های نقدینگی

This paper examines the continuous time optimal consumption and portfolio choice of an investor having an initial wealth endowment and an uncertain stream of income from non-traded assets. The income stream is not spanned by traded assets and the investor is not allowed to borrow against future income, so the financial market is incomplete. We solve the corresponding stochastic control problem numerically with the Markov chain approximation method, prove convergence of the method, and study the optimal policies. In particular, we find that the implicit value the agent attaches to an uncertain income stream typically is much smaller in this incomplete market than it is in the otherwise identical complete market. Our results suggest that this is mainly due to the presence of liquidity constraints.

The consumption/investment choice of a price-taking investor is a classical problem of financial economics. In two pioneering papers, Merton 1969 and Merton 1971 introduced stochastic control techniques to analyze the continuous-time version of the problem for an investor with an additively time-separable utility function. In particular, Merton studied the case where the investor has access to a complete financial market in which risky asset prices follow geometric Brownian motions and the investor's utility function for consumption is of the constant relative risk aversion type. He was able to solve analytically the Hamilton–Jacobi–Bellman (HJB) equation associated with the problem and hence to obtain closed-form expressions for the optimal control policies in feedback form, both for a finite horizon and an infinite horizon. One of many interesting generalizations of Merton's setting appears when the investor besides having an initial endowment of wealth also receives a stream of income throughout her planning horizon. Merton (1971, Section 7) stated that the optimal policies when the agent has a deterministic stream of income are as if the agent has no income stream but instead adds the capitalized lifetime income flow discounted at the risk-free rate to her initial wealth. However, it is easy to show, see e.g. He and Pagès (1993, Example 1), that under the policies derived this way the wealth process may go below zero. Due to moral hazard and adverse selection problems, it may be impossible for the investor to borrow against future income, so that the investor can only choose her consumption/investment policy among those that keep her financial wealth non-negative. He and Pagès (1993) study a model where the income rate is spanned, so that the only source of incompleteness is that liquid wealth has to stay non-negative. Using the martingale techniques of Cox and Huang (1989) they find that the presence of liquidity constraints has a smoothing effect on the optimal consumption across time. If the investor expects her income to rise, she will increase her consumption at a smaller rate than if she was not subjected to liquidity constraints. In a similar set-up, El Karoui and Jeanblanc-Picqué (1996) demonstrate that the optimal trading strategy is to invest part of the wealth in the strategy which is optimal in the corresponding unconstrained case, and the remainder in an American put option written on the optimal wealth process in the unconstrained case.1 They also derive a formula linking the optimal consumption rate to current wealth and current income, and they show that for zero wealth the optimal consumption rate is a smaller fraction of current income in the liquidity constrained case than in the unconstrained case. Maintaining the liquidity constraints of He and Pagès, but dropping the spanning assumption, Duffie and Zariphopoulou (1993) study an infinite horizon model with a single risky asset whose price follows a geometric Brownian motion and a non-negative income rate given by an Itô process driven by a Brownian motion imperfectly correlated with the risky asset price. In this general setting we cannot be sure that the value function, also known as the indirect utility function, is a solution of the associated HJB equation. In fact the value function may not even be smooth. Duffie and Zariphopoulou are able to show that the value function is the unique constrained viscosity solution in the class of concave functions of the HJB equation. The model of Duffie and Zariphopoulou is specialized in Duffie et al. (1997), henceforth abbreviated DFSZ. For power utility of consumption they reduce the control problem from a two-variable (wealth and income) to a one-variable (wealth divided by income) problem and show that the HJB equation for the reduced problem has a unique smooth solution. The optimal policies and the value function of the original problem can easily be restored from the solution to the reduced problem. With analytical derivations DFSZ are able to find some characteristics of the solution, but they cannot solve the problem completely. In an almost identical model Koo (1998) expresses the optimal policies in terms of the current liquid wealth and a measure of the agent's implicit value of the future uncertain income stream and is able to analytically derive some general properties of these policies. In Section 2 below, we shall review the main findings of DFSZ and Koo, and we introduce an intuitively more appealing measure of the implicit value of future income and relate it to Koo's measure. For other continuous-time models with stochastic income, see Andersson et al. (1995), Cuoco (1997), Detemple and Serrat (1997), and Svensson and Werner (1993). The main contributions of this paper are to demonstrate how the reduced problem of DFSZ can be solved by a relatively simple converging numerical scheme and to study the properties of optimal policies in the set-up of DFSZ in detail. The numerical method adopted is the Markov chain approximation approach which basically approximates the continuous time, continuous state stochastic control problem with a discrete time, discrete state stochastic control problem that is easily solved numerically. The Markov chain approximation approach is described in Kushner (1990) and more detailed in Kushner and Dupuis (1992). See also Fleming and Soner (1993, Chap. IX). The method has previously been applied to consumption/portfolio problems by Fitzpatrick and Fleming (1991) and Hindy et al. (1997). We contrast the numerically computed value function and optimal controls to the complete market case where the income rate is spanned by traded assets and the investor is not liquidity constrained. In particular, we find that the implicit value the investor attaches to the uncertain income stream is much smaller in the non-spanned, liquidity constrained case than in the complete market case, even for high ratios of initial wealth to initial income. We find that this implicit value of income is very insensitive to the correlation between changes in the risky asset price and changes in the income rate. Since a perfectly positive correlation corresponds to spanning, this suggests that the large difference between the complete markets case and the non-spanned, liquidity constrained case is to be attributed to the liquidity constraints. We study the sensitivity of both the optimal policies and the implicit value of the income stream with respect to various parameters. Among other things we find that, ceteris paribus, liquidity constraints as modeled in this paper are most restrictive for agents with a low financial wealth relative to income, an income rate positively correlated with changes in the risky asset price, and a high time preference for consumption. While this paper apparently contains the first quantitative study of the properties of the optimal consumption and portfolio policies in a continuous-time set-up with undiversifiable income risk and liquidity constraints, related discrete-time models have been considered by other authors. Koo (1995) also has a non-spanned income process and a liquidity constraint, but in a discrete-time set-up the liquidity constraint explicitly implies bounds on the risky investment in each period, which is not the case with continuous trading. Furthermore, Koo imposes even tighter bounds on the investment in the risky asset than those implied by his liquidity constraint. Hence, it is difficult to evaluate the implications of the liquidity constraint from his numerical results. Koo only considers the case where the shocks to the income rate process and the shocks to the risky asset price are uncorrelated, while we allow for a non-zero correlation and provide a detailed comparative statics analysis. For other discrete-time models of optimal consumption and investment choice with income, see, e.g., Cocco et al. (1998), Deaton (1991), and Heaton and Lucas (1997). The outline of the rest of the paper is as follows. The problem is formalized in Section 2 which also reviews the analytical findings of DFSZ and discusses two implicit measures of the value of an uncertain non-spanned stream of income. In Section 3, we implement the Markov chain approximation method and prove its convergence. Numerical results are presented and discussed in Section 4. In Section 5, we study the sensitivity of the results with respect to selected parameters. Finally, Section 6 briefly summarizes the paper.

In this paper we have studied the properties of the optimal consumption and investment policies of a liquidity constrained investor with an uncertain, non-spanned income process in a continuous-time set-up. The underlying optimization problem was solved numerically with the Markov chain approximation approach and the numerical solution was shown to converge to the true solution of the problem. We have found that the implicit value the investor associates with the entire income process is much smaller in the presence of liquidity constraints and undiversifiable income risk than without such imperfections. Our results suggest that this is mainly due to the presence of liquidity constraints.